Fields in Nonaffine Bundles. I. The general bitensorially covariant differentiation procedure

TL;DR

This paper extends covariant differentiation to fields in nonaffine bundles with nonlinear fibres, requiring an intrinsic connection on the fibre space, and introduces a new connector based on fibre-tangent vectors.

Contribution

It develops a general differentiation procedure for nonaffine bundle fields, incorporating an intrinsic fibre connection and a new connector construction.

Findings

Generalized covariant differentiation for nonaffine bundles

Introduced a fibre-group invariant connection on the fibre space

Applicable to gauged harmonic mappings in future work

Abstract

The standard covariant differentiation procedure for fields in vector bundles is generalised so as to be applicable to fields in general nonaffine bundles in which the fibres may have an arbitrary nonlinear structure. In addition to the usual requirement that the base space should be flat or endowed with its own linear connection, and that there should be an ordinary gauge connection on the bundle, it is necessary to require also that there should be an intrinsic, bundle-group invariant connection on the fibre space. The procedure is based on the use of an appropriate primary-field (i.e. section) independent connector that is constructed in terms of the natural fibre-tangent-vector realisation of the gauge connection. The application to gauged harmonic mappings will be described in a following article.